[orrinward]

"arbitrage scheme in an exchange, where the quotes might vary over time" - The nature of arbitrage is that the profit opportunity is available at a single moment.

What I interpret your explanation as is more akin to trading - Buying backs at a time and waiting for odds to fluctuate at an exchange for a sale.

Although there is no (simple) way of doing a pure arb between a bookie/exchange, all the popular exchanges expose existing liquidity in a market and you can quite safely reduce the risk by double checking available liquidity at the exchange in the seconds before placing your back bet.